On the geometry of a (q + 1)-arc of PG(3,q), q even

Abstract

In PG(3,q), q=2n, n≥3, let Ah,q={(1,t,t2h,t2h+1)|t∈Fq}∪{(0,0,0,1)}, with gcd⁡(n,h)=1, be a (q+1)-arc and let Gh≃PGL(2,q) be the stabilizer of Ah,q in PGL(4,q). The Gh-orbits on points, lines and planes of PG(3,q), together with the point-plane incidence matrix with respect to the Gh-orbits on points and planes of PG(3,q) are determined. The point-line incidence matrix with respect to the G1-orbits on points and lines of PG(3,q) is also considered. In particular, for a line ℓ belonging to a given line G1-orbit, say L, the point G1-orbit distribution of ℓ is either explicitly computed or it is shown to depend on the number of elements x in Fq (or in a subset of Fq) such that Trq|2(g(x))=0, where g is an Fq-map determined by L.